in a motion acceleration directly proportional to time . find velocity and displacement after time t second
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Answer:
For the sake of accuracy, this section should be entitled "One dimensional equations of motion for constant acceleration". Given that such a title would be a stylistic nightmare, let me begin this section with the following qualification. These equations of motion are valid only when acceleration is constant and motion is constrained to a straight line.
Given that we live in a three dimensional universe in which the only constant is change, you may be tempted to dismiss this section outright. It would be correct to say that no object has ever traveled in a straight line with a constant acceleration anywhere in the universe at any time — not today, not yesterday, not tomorrow, not five billion years ago, not thirty billion years in the future, never. This I can say with absolute metaphysical certainty.
So what good is this section then? Well, in many instances, it is useful to assume that an object did or will travel along a path that is essentially straight and with an acceleration that is nearly constant; that is, any deviation from the ideal motion can be essentially ignored. Motion along a curved path may be considered effectively one-dimensional if there is only one degree of freedom for the objects involved. A road might twist and turn and explore all sorts of directions, but the cars driving on it have only one degree of freedom — the freedom to drive in one direction or the opposite direction. (You can't drive diagonally on a road and hope to stay on it for long.) In this regard, it is not unlike motion restricted to a straight line. Approximating real situations with models based on ideal situations is not considered cheating. This is the way things get done in physics. It is such a useful technique that we will use it over and over again.
Our goal in this section then, is to derive new equations that can be used to describe the motion of an object in terms of its three kinematic variables: velocity (v), position (s), and time (t). There are three ways to pair them up: velocity-time, position-time, and velocity-position. In this order, they are also often called the first, second, and third equations of motion, but there is no compelling reason to learn these names.
Since we are dealing with motion in a straight line, direction will be indicated by sign — positive quantities point one way, while negative quantities point the opposite way. Determining which direction is positive and which is negative is entirely arbitrary. The laws of physics are isotropic; that is, they are independent of the orientation of the coordinate system. Some problems are easier to understand and solve, however, when one direction is chosen positive over another. As long as you are consistent within a problem, it doesn't matter.
velocity-time
The relation between velocity and time is a simple one during uniformly accelerated, straight-line motion. The longer the acceleration, the greater the change in velocity. Change in velocity is directly proportional to time when acceleration is constant. If velocity increases by a certain amount in a certain time, it should increase by twice that amount in twice the time. If an object already started with a certain velocity, then its new velocity would be the old velocity plus this change. You ought to be able to see the equation in your mind's eye already.
This is the easiest of the three equations to derive using algebra. Start from the definition of acceleration.
a = ∆v
∆t
Expand ∆v to v − v0 and condense ∆t to t.
a = v − v0
t
Then solve for v as a function of t.
v = v0 + at [1]